Linear Reformulation Research Articles

A novel set-induced robust optimization model under correlated uncertainty: a comparative theoretical and computational study

In practical scenarios, optimization problems frequently involve uncertain parameters that exhibit correlated fluctuations. The widely adopted set-induced robust optimization approach is employed to address correlated uncertainty in environments with limited data. This approach focuses on deriving a deterministic reformulation of the original uncertain optimization problem, ensuring that the constructed solution remains feasible for all potential realizations of input parameters within a predefined uncertainty set. However, this approach often yields overly conservative solutions. In this study, an innovative uncertainty set is developed to mitigate the over-conservatism associated with the set-induced robust optimization approach under correlated uncertainty. Moreover, from the proposed description of data uncertainty, a robust linear reformulation of the original uncertain optimization problem is established to strike a balance between conservativeness and optimality. Computational studies exploring production planning optimization programs demonstrate the superior performance of the proposed robust model over other alternatives, considering various levels of correlation among uncertain parameters.

On exact and inexact RLT and SDP-RLT relaxations of quadratic programs with box constraints

Quadratic programs with box constraints involve minimizing a possibly nonconvex quadratic function subject to lower and upper bounds on each variable. This is a well-known NP-hard problem that frequently arises in various applications. We focus on two convex relaxations, namely the reformulation–linearization technique (RLT) relaxation and the SDP-RLT relaxation obtained by combining the Shor relaxation with the RLT relaxation. Both relaxations yield lower bounds on the optimal value of a quadratic program with box constraints. We show that each component of each vertex of the RLT relaxation lies in the set {0,12,1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{0,\\frac{1}{2},1\\}$$\\end{document}. We present complete algebraic descriptions of the set of instances that admit exact RLT relaxations as well as those that admit exact SDP-RLT relaxations. We show that our descriptions can be converted into algorithms for efficiently constructing instances with (1) exact RLT relaxations, (2) inexact RLT relaxations, (3) exact SDP-RLT relaxations, and (4) exact SDP-RLT but inexact RLT relaxations. Our preliminary computational experiments illustrate that our algorithms are capable of generating computationally challenging instances for state-of-the-art solvers.

Drone-Delivery Network for Opioid Overdose: Nonlinear Integer Queueing-Optimization Models and Methods

This study proposes an emergency network design model that uses a fleet of drones to deliver naloxone in response to opioid overdoses. The network is represented as a collection of M/G/K queueing systems with variable capacity and service time modelled as a decision-dependent random variable. The model is a complex queuing-based optimization problem which locates drone bases and dispatches drones to opioid incidents. The authors devise an efficient solution framework in which they linearize the multiple nonlinearities (fractional, polynomial, exponential, factorial terms), derive an equivalent mixed-integer linear reformulation, and design an outer approximation branch-and-cut algorithm. The authors demonstrate the generalizablity of the approach to any problem minimizing the response time of M/G/K queueing systems with unknown capacity. Tests on Virginia Beach data reveal that drones can decrease response time by 82%, increase survival chance by more than 273%, save up to 33 additional lives annually, and provide up to 279 additional quality-adjusted life years.

Solving a Multimodal Routing Problem with Pickup and Delivery Time Windows under LR Triangular Fuzzy Capacity Constraints

This study models a container routing problem using multimodal transportation to improve its economy, timeliness, and reliability. Pickup and delivery time windows are simultaneously formulated in optimization to provide the shipper and the receiver with time-efficient services, in which early pickup and delayed delivery can be avoided, and nonlinear storage periods at the origin and the destination can be minimized. Furthermore, the capacity uncertainty of the multimodal network is incorporated into the advanced routing to enhance its reliability in practical transportation. The LR triangular fuzzy number is adopted to model the capacity uncertainty, in which its spread ratio is defined to measure the uncertainty level of the fuzzy capacity. Due to the nonlinearity introduced by the time windows and the fuzziness from the network capacity, this study establishes a fuzzy nonlinear optimization model for optimization problem. A chance-constrained linear reformulation equivalent to the proposed model is then generated based on the credibility measure, which makes the global optimum solution attainable by using Lingo software. A numerical case verification demonstrates that the proposed model can effectively solve the problem. The case analysis points out that the formulation of pickup and delivery time windows can improve the timeliness of the entire transportation process and help to achieve on-time transportation. Furthermore, improving the confidence level and the uncertainty level increases the total costs of the optimal route. Therefore, the shipper and the receiver must prepare more transportation budget to improve reliability and address the increasing uncertainty level. Further analysis draws some insights to help the shipper, receiver, and multimodal transport operator to organize a reliable and cost-efficient multimodal transportation under capacity uncertainty through confidence level balance and transportation service and transfer service selection.

Omnichannel product selection and shelf space planning optimization

As the retail industry is evolving toward an omnichannel paradigm, we study a product selection and shelf space planning problem faced by omnichannel retailers. We aim to maximize aggregated profits across channels by determining the optimal product offerings for the offline and online channels while optimizing the allocation of limited shelf space to these products in brick-and-mortar stores. Furthermore, the showroom effect of physical stores on online sales is also taken into account. We propose a corresponding model that jointly optimizes these decisions. To solve the nonlinear nonconvex model of practical scale, we first reformulate it into a mixed integer quadratic programming by exploring the structural properties. Next, we use the reformulation linearization technique to further improve the computational speed. Our numerical studies validate the efficiency of the proposed solution approach. In addition, we derive some managerial insights, including that retailers should adopt joint optimization and consider how the involved effects and parameters may influence their profits and decisions.

Decentralized exam timetabling: A solution for conducting exams during pandemics

Traditionally, exams were scheduled into designated timeslots within a centralized campus. However, the outbreak of COVID-19 disrupted this practice, forcing universities to prioritize the safety of individuals during exams. In response, a shift to online assessments occurred, bringing forth new challenges such as academic dishonesty and inadequate infrastructure. This study presents a novel approach to decentralized in-person exams for geographically dispersed students. A mixed-integer non-linear programming model is developed, followed by a mixed-integer linear reformulation, to optimally determine the scheduling of exams, test center locations, and student allocation, while addressing the incorporation of both hard and soft constraints. Given the complexity of the problem, we have implemented meta-heuristic algorithms to solve larger instances, including genetic and simulated-annealing algorithms. Additionally, we have devised a hybrid algorithm that combines the strengths of population-based and local-search algorithms. For evaluation purposes, we used real data collected from our department during the fall semester of 2022. The results demonstrate the superior performance of the hybrid algorithm compared to others, in terms of both effectiveness and adaptability. It remains consistent across different scenarios, effectively addressing real-world complexities. Finally, a comparison between centralized and decentralized scenarios underscores the advantages of the latter approach during the pandemic.

Active Distribution Network Expansion Planning With Dynamic Thermal Rating of Underground Cables and Transformers

The rapid development of renewable distributed generation in active distribution networks (ADNs) imposes an increasing burden on the transfer capability of the ADNs, bringing new challenges to the distribution network expansion planning (DNEP) problem. Dynamic thermal rating (DTR), which evaluates the equipment rating based on the actual weather conditions and equipment thermal states, can enhance the equipment transfer capability to support the integration of renewable distributed generation. In this paper, we propose a DNEP model of ADNs incorporating the DTR of cables and transformers, given that underground cable feeder is preferred in the rapid urbanization trends. Then, we derive a linear reformulation of the original DNEP model and propose a modified Benders decomposition (MBD) algorithm to solve the DNEP model. To select more effective representative day scenarios, we propose a cost-based clustering method for representative day selection applicable to solving the DNEP model. Case studies based on the IEEE 33-node system and the PG&E 69-node system show that the implementation of DTR saves 14.7% and 15.1% of the investment costs of the two systems respectively. The effectiveness of the MBD algorithm and the cost-based clustering method is also verified by the case studies.

Distributionally robust joint chance-constrained programming: Wasserstein metric and second-order moment constraints

In this paper, we propose a new approximate linear reformulation for distributionally robust joint chance programming with Wasserstein ambiguity sets and an efficient solution approach based on Benders decomposition. To provide a convex approximation to the distributionally robust chance constraint, we use the worst-case conditional value-at-risk constrained program. In addition, we derive an approach for distributionally robust joint chance programming with a hybrid ambiguity set that combines a Wasserstein ball with second-order moment constraints. This approach, which allows injecting domain knowledge into a Wasserstein ambiguity set and thus allows for less conservative solutions, has not been considered before. We propose two formulations of this problem, namely a semidefinite programming and a computationally favorable second-order cone programming formulation. The models and algorithms proposed in this paper are evaluated through computational experiments demonstrating their computational efficiency. In particular, the Benders decomposition algorithm is shown to be more than an order of magnitude faster than a standard solver allowing for the solution of large instances in a relatively short time.

Data-driven joint optimization of maintenance and spare parts provisioning for deep space habitats: A stochastic programming approach

The utilization of sensor data to make informed decisions on maintenance schedule and spare parts provisioning will be determinant to the development of self-aware habitable facilities–so called SmartHabs–in deep space regions. To address this novel and challenging problem, this work presents a chance constrained stochastic mixed integer program to jointly optimize maintenance and spare parts provisioning for ensuring reliable and cost-efficient operations of SmartHabs. A SmartHab is modeled as a multi-unit system with non-identical components that randomly degrade over time. The proposed optimization model is driven by telemetry sensor data as the uncertainty of the remaining lifetime of the components is estimated by data-driven prognostic algorithms. The model further incorporates characteristic operational conditions of deep space habitats such as a long planning horizon that requires multiple maintenance decisions for each component, maintenance task assignment to humans and robots, long lead times between resupply trips, and high penalties due to stock-outs of spare parts. The formulation introduces chance constraints restricting the unavailability of critical components and the number of corrective repairs with high probability under the uncertainty of component failures. The nonlinearities induced by the chance constraints are circumvented by the derivation of two linearization methods based on scenario representation and safe approximations, resulting in their tractable mixed integer linear reformulations. The proposed model is validated by simulations, showing reliable and effective plans against benchmark approaches while evidencing to be computationally scalable as the number of components increases.

An efficient adaptive large neighborhood search algorithm based on heuristics and reformulations for the generalized quadratic assignment problem

Operations Research (OR) analytics play a crucial role in optimizing decision-making processes for real-world problems. The assignment problem, with applications in supply chains, healthcare logistics, and production scheduling, represents a prominent optimization challenge. This paper focuses on addressing the Generalized Quadratic Assignment Problem (GQAP), a well-known NP-hard combinatorial optimization problem. To tackle the GQAP, we propose an OR analytical approach that incorporates efficient relaxations, reformulations, heuristics, and a metaheuristic algorithm. Initially, we employ the Reformulation Linearization Technique (RLT) to generate various linear relaxation models, carefully selecting the most efficient ones. Building upon this foundation, we introduce a robust reformulation based on Benders Decomposition (BD), which serves as the basis for an iterative optimization algorithm applied to the GQAP. Furthermore, we develop a constructive heuristic algorithm to identify near-optimal solutions, followed by an enhancement utilizing an Adaptive Large Neighborhood Search (ALNS) metaheuristic algorithm. This ALNS algorithm is enhanced through the integration of a tabu list derived from Tabu Search (TS) and a decision rule inspired by Simulated Annealing (SA). To validate our approach and evaluate its performance, we conduct a comparative analysis against state-of-the-art algorithms documented in the literature. This comparison confirms the significant improvements achieved in terms of solution quality and computational efficiency through our proposed methodology. These advancements contribute to the state of the art in solving the GQAP and hold the potential to enhance decision-making processes in a wide range of domains. Our methodology demonstrates remarkable improvements in solution quality and computational efficiency when compared to existing approaches, as evidenced by the comparative results with state-of-the-art algorithms. The potential implications of our research extend to optimizing decision-making processes in diverse fields, rendering it highly relevant and impactful.

Distributed and Collective Intelligence for Computation Offloading in Aerial Edge Networks

Unmanned aerial vehicles (UAVs) with integrated computing platforms can be used to provide computing offloading services for ground user equipments (UEs) with limited local computing capabilities, especially in remote areas. In this paper, we focus on the task offloading in an aerial edge network (AEN) assisted by a UAV. We aim at minimizing the sum energy consumption of all UEs by the joint optimization of the task offloading decisions and the UAV position under the constraints of the latency and the total energy of UAV. The formulated optimization problem is a mixed-integer nonconvex problem and involves coupling of many optimization variables. To address this challenge, we first transform the original optimization problem into a linear convex optimization problem via reformulation linearization technology, and then the alternating direction method of multipliers (ADMM) algorithm is proposed to achieve the approximate optimal solution. Numerical results confirm that the proposed ADMM algorithm can effectively reduce the total of energy consumption of UEs and ensure the continuous operation of the UEs.

Efficient linear reformulations for binary polynomial optimization problems

We consider unconstrained polynomial minimization problems with binary variables (BPO). These problems can be easily linearized, i.e., reformulated into a MILP in a higher dimensional space. Several linearizations are possible for a given BPO, depending on how each monomial is decomposed and replaced by additional variables and constraints. We focus on finding efficient linearizations that maximize the continuous relaxation bound of the resulting MILP. For this purpose, we introduce the notion of linearization patterns that allow us to model and enumerate the possible decompositions of a degree-d monomial. The assignment of a unique pattern to each monomial of BPO results in a reformulation of BPO into a MILP. Our method, called MaxBound, amounts to searching for an optimal association between monomials and patterns in the sense that it leads to a MILP with the best continuous relaxation bound. We show that this process can be formulated as a MILP which we denote by (MB˜). We further highlight domination properties among the patterns that allow us to discard the dominated patterns and to decrease the size of (MB˜). Another effect of these domination properties is that it now makes sense to search for a reformulation that requires as few additional variables as possible, based only on the non-dominated patterns. We call this reformulation method ND-MinVar and again show that it can be found by solving another MILP. We make an experimental study on degree 4 polynomials that compares the results of both methods and shows the advantages and disadvantages of each.

Joint Task Offloading and Resource Allocation for Vehicular Edge Computing Based on V2I and V2V Modes

In an internet of vehicle (IoV) scenario, vehicular edge computing (VEC) exploits the computing capabilities of the vehicles and roadside unit (RSU) to enhance the task processing capabilities of the vehicles. Resource management is essential to the performance improvement of the VEC system. In this paper, we propose a joint task offloading and resource allocation scheme to minimize the total task processing delay of all the vehicles through task scheduling, channel allocation, and computing resource allocation for the vehicles and RSU. Different from the existing works, our scheme: 1) considers task diversity by profiling the tasks of the vehicles by multiple attributes including data size, computation amount, delay tolerance, and task type; 2) considers vehicle classification by dividing the vehicles into 4 sets according to whether they have task offloading requirements or provide task processing services; 3) considers task processing flexibility by deciding for each vehicle to process its tasks locally, to offload the tasks to the RSU via V2I (Vehicle to Infrastructure) connections, or to the other vehicles via V2V (Vehicle to Vehicle) connections. An algorithm based on the Generalized Benders Decomposition (GBD) and Reformulation Linearization (RL) methods is designed to optimally solve the optimization problem. A heuristic algorithm is also designed to provide the sub-optimal solution with low computational complexity. We analyze the convergence and complexity of the proposed algorithms and conduct extensive simulations in 6 scenarios. The simulation results demonstrate the superiority of our scheme in comparison with 4 other schemes.

The follower competitive facility location problem under the nested logit choice rule

Recently, more realistic customer choice rules based on the Multinomial Logit have been proposed for the follower competitive facility location problem, together with efficient exact solution methods based on cut generation approaches. We address this problem in the case in which a new chain, the entrant, decides to locate some stores in a market where there are already incumbent chains offering substitute products, which differ from the entrant's product in secondary features. The entrant's stores are very similar to each other, and this is also the case for the incumbent chains. In this case, for a given customer, purchases made at the stores of one of the chains are correlated, given the similitude between these stores. The Multinomial Logit rule does not capture this correlation. We propose using a Nested Logit rule which does capture this correlation and, in addition, represents a sequential decision process by the customer, who first chooses the chain and then the store. We prove the concavity of the objective function in the problem and find exact solutions using Branch and Cut over a generalized linear reformulation of the problem and four different types of cuts: submodular and an improvement of these, outer-approximation, and a set of new cuts. Our results show locations that fit better the intuition and practice. Furthermore, our computational times improve in many cases upon the known results for similar problems.

Multi-time scale stochastic production simulation under VHVDC long-term contract trading electricity quantity constraint

A two-layer multi-time scale stochastic production simulation framework is constructed to account for the long-term contract electricity quantity of ultra-high voltage direct current (UHVDC) transmission. On the upper layer, based on the characteristics of load demand and renewable energy output extracted from the historical operating data, monthly and daily production simulation models are carried out considering the seasonal characteristics of hydropower during a high-water period and low-water period to optimize the distribution of contract electric quantity sending through UHVDC transmission in the target year or month. According to the DC transmission electric quantity optimized by the daily production simulation in the upper layer, together with the forecast scenario, the lower layer of the framework provides the optimization of day-ahead scheduling and intra-day rolling dispatch in the implementation process. The day-ahead dispatch optimization makes full use of the adjustment capability of transmission and optimizes the DC transmission electric quantity correction. Its compensation is based on the result of the daily production simulation, then the correction will be returned to the upper layer to restart the optimization of the remaining UHVDC contract electric quantity of the subsequent period and its distribution plan. Combined with the day-ahead DC transmission plan, the intra-day rolling optimization is carried out to adjust the output of the unit using more accurate forecasting scenarios. The distributionally robust optimization model is used in the lower layer to convert an uncertain problem into a deterministic quadratically constrained quadratic programming (QCQP) problem according to the form of an uncertain distribution set. Then the QCQP problem is further converted into a linear programming (LP) problem by using the reformulation linearization technique (RLT). A test system with the energy composition and distribution referring to a real provincial power grid in northwest China is established for verification. The results show that the proposed method can effectively improve the economics of system operation and the accommodation of renewable energy based on ensuring security.

The Value of Randomized Strategies in Distributionally Robust Risk-Averse Network Interdiction Problems

Conditional value at risk (CVaR) is widely used to account for the preferences of a risk-averse agent in extreme loss scenarios. To study the effectiveness of randomization in interdiction problems with an interdictor that is both risk- and ambiguity-averse, we introduce a distributionally robust maximum flow network interdiction problem in which the interdictor randomizes over the feasible interdiction plans in order to minimize the worst case CVaR of the maximum flow with respect to both the unknown distribution of the capacity of the arcs and the interdictor’s own randomized strategy. Using the size of the uncertainty set, we control the degree of conservatism in the model and reformulate the interdictor’s distributionally robust optimization problem as a bilinear optimization problem. For solving this problem to any given optimality level, we devise a spatial branch-and-bound algorithm that uses the McCormick inequalities and reduced reformulation linearization technique to obtain a convex relaxation of the problem. We also develop a column-generation algorithm to identify the optimal support of the convex relaxation, which is then used in the coordinate descent algorithm to determine the upper bounds. The efficiency and convergence of the spatial branch-and-bound algorithm is established in numerical experiments. Further, our numerical experiments show that randomized strategies can have significantly better performance than optimal deterministic ones.History: Accepted by David Alderson, Area Editor for Network Optimization: Algorithms & Applications.Funding: The first author’s research is supported by a Group for Research in Decision Analysis postdoctoral fellowship and Fonds de Recherche du Québec–Nature et Technologies postdoctoral research scholarship [Grants 275296 and 301065]. The second author acknowledges support from the Canadian Natural Sciences and Engineering Research Council and the Canada Research Chair program [Grants RGPIN-2016-05208, 492997-2016, and 950-230057].Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2022.1257 . The data file and codes are posted on GitHub ( https://github.com/Utsav19/Value-of-Randomization ).

Multi-period distribution network design with boundedly rational customers for the service-oriented manufacturing supply chain: a 4PL perspective

With the service level playing an increasingly essential role in the service-oriented manufacturing (SOM) supply chain, the distribution network design can be heavily affected by the customer behaviour based on their satisfaction of services. In this paper, we consider the service level for service time and delivery quantity separately. A novel mixed integer non-linear programming model is proposed to design the multi-period distribution network from a fourth-party logistics (4PL) perspective. The customer satisfaction based on prospect theory is maximised while considering the investment budget and the service level. A scenario-based linear reformulation is proposed to find the optimal solution when the problem scale is small. For a large-scale problem, we propose an individual-driven Q-learning based memetic particle swarm optimisation algorithm. Numerical experiments are conducted to demonstrate the effectiveness and efficiency of the proposed algorithm. Furthermore, the impact of service modes, different customer behaviour, and customer satisfaction evaluation periods on distribution network is investigated. We find that the length of evaluation periods leads to differences in customer satisfaction due to different perceptions of ‘small loss’ and ‘big gain’ by boundedly rational customers.

Joint Task Offloading and Resource Allocation for Multihop Industrial Internet of Things

Task offloading in edge computing is important for the Industrial Internet of Things (IIoT) to implement computation-intensive applications in real time. However, achieving efficient task offloading in IIoT is very challenging due to the limited computing resources of IIoT devices, the coupling of computing and communication resources, and the unreliability in multihop wireless transmission. In this article, we construct a link model by considering the influence of unreliable links in multihop transmission to reveal the relationship between reliability and transmission delay. Then, a nonconvex optimization problem that minimizes task processing delay is formulated, and task offloading is decided by considering transmission path selection, bandwidth allocation, and computational resource allocation. To solve this problem, an algorithm based on the alternating direction method of multipliers (ADMM) is designed using auxiliary variables and reformulation linearization technology (RLT). The simulation results show that our proposed algorithm can fully utilize the computing power of the edge server and reduce the task processing delay. Compared with the centralized algorithms, the performance of the proposed scheme is only 1% worse, but the calculation time can be reduced by 40%.

Tight Neural Network Verification via Semidefinite Relaxations and Linear Reformulations

We present a novel semidefinite programming (SDP) relaxation that enables tight and efficient verification of neural networks. The tightness is achieved by combining SDP relaxations with valid linear cuts, constructed by using the reformulation-linearisation technique (RLT). The computational efficiency results from a layerwise SDP formulation and an iterative algorithm for incrementally adding RLT-generated linear cuts to the verification formulation. The layer RLT-SDP relaxation here presented is shown to produce the tightest SDP relaxation for ReLU neural networks available in the literature. We report experimental results based on MNIST neural networks showing that the method outperforms the state-of-the-art methods while maintaining acceptable computational overheads. For networks of approximately 10k nodes (1k, respectively), the proposed method achieved an improvement in the ratio of certified robustness cases from 0% to 82% (from 35% to 70%, respectively).

Strengthening a linear reformulation of the 0-1 cubic knapsack problem via variable reordering

Strengthening a linear reformulation of the 0-1 cubic knapsack problem via variable reordering